Nambu Structures And Associated Bialgebroids

Abstract

This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order $n>2$) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.